OFFSET
1,1
COMMENTS
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
An integer partition is 2-vertex-connected if the prime factorizations of the parts form a connected hypergraph that is still connected if any single prime number is divided out of all the parts (and any parts then equal to 1 are removed).
LINKS
EXAMPLE
The sequence of all 2-vertex-connected integer partitions begins: (1), (6), (10), (12), (6,2), (14), (15), (18), (6,3), (20), (21), (22), (10,2), (24), (6,4), (26), (28), (12,2), (30), (6,2,2), (14,2), (33), (34), (35), (36), (38), (39), (6,6), (40), (42), (18,2), (12,3), (44), (6,3,2), (45), (46).
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]], 2], And[OrderedQ[#], UnsameQ@@#, Length[Intersection@@s[[#]]]>0]&]}, If[c=={}, s, csm[Sort[Append[Delete[s, List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
vertConn[y_]:=If[Length[csm[primeMS/@y]]!=1, 0, Min@@Length/@Select[Subsets[Union@@primeMS/@y], Function[del, Length[csm[DeleteCases[DeleteCases[primeMS/@y, Alternatives@@del, {2}], {}]]]!=1]]]
Select[Range[100], vertConn[primeMS[#]]>1&]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 05 2018
STATUS
approved